group

Hello

1. G is a group of order , with p prime number and n natural number, which acts on a finite set of cardinal X not divisible by p. Show that there is some element x in X such that gx = x for all g in G.

2. If p and q are primes with p <q. Prove that every group G of order pq has only one subgroup of order q normal in G. If q is not congruent to 1 modulo p, show that G is abelian and cyclic.

(one can also show directly that for a generator h of H, and generator k of K, the element hk has order pq = lcm(p,q)).

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to show that a group of order pq where p divides q-1 need not be abelian (let alone cyclic), let p = 2, and q be any odd prime, and consider the dihedral group of order 2q (for q = 3, this is isomorphic to S3).

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in all fairness, a proof using the sylow theorems *would* be shorter in showing K is normal in G. but that would not tell one how to show that HK is abelian.